Acceleration of cosmic rays by double shock waves in galaxy clusters: application to radio relics
AAstronomy & Astrophysics manuscript no. paper © ESO 2021March 1, 2021
Acceleration of cosmic rays by double shock waves in galaxyclusters: application to radio relics
Grazyna Siemieniec–Ozieblo and Mariia Bilinska
Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30–244 Kraków, Polande-mail: [email protected]; [email protected]
Received < date > / Accepted < date > ABSTRACT
Context.
Radio relics in galaxy clusters are known to be good laboratories for verification of the applicability of the di ff usive shock ac-celeration (DSA) model in its canonical version. The need for such verification stems from the inconsistencies in the shock propertiesresulting from radio observations compared to X-ray observations. Aims.
In this article we aim to explore how the presence of a second shock in the vicinity of a relic modifies the spectrum of acceleratedelectrons and decipher which of the involved parameters can have a significant impact on their shape.
Methods.
We analytically studied DSA of cosmic rays in two stationary shocks aiming to investigate the change of the distributionfunction. The latter eventually leads to spectrum slope deviations visible in di ff erent observations and simulations that do not appearto be explained by the case wherein cosmic rays interact with a single shock wave. Results.
We obtain a complex distribution function f ( x , p ) depending on many parameters (distance between two shocks, compressionratios, spatial di ff usion coe ffi cients, injection value, etc.). This function reveals modifications that occur because of the coupledacceleration in both shocks. Apparently, deviations in the particle spectrum from the pure power law depend on a few parameterssuch as Q / Q , κ /κ , r / r , and L. Although we do not verify this idea by taking a particular cluster as an example, we demonstrate apotential cause of spectral disturbances in radio relics. In general terms, our findings appear to correlate with results from the literaturewhen the distance between the shocks is of the order of the width of a radio relic and κ /κ ∝ Key words.
Galaxies: clusters: general — acceleration of particles — cosmic rays — shock waves — methods: analytical
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1. Introduction
According to the large-scale structure (LSS) formationparadigm, the evolution of galaxy clusters is a hierarchical pro-cess consisting of continuous mergers and accretion of smallersystems. Both acts of merging and accretion are accompaniedby the appearance of shock waves. In the case of galaxy clus-ters, these are shocks moving at nonrelativistic velocities. Whenshocks are moving, their huge kinetic energy is distributed intodi ff erent channels (e.g., Vazza et al. 2009). Although the mainrole of shocks is to heat plasma in intra-cluster medium (ICM),clusters are also energy reservoirs for a number of other nonther-mal phenomena leading on the one hand to particle accelerationand on the other to amplification of magnetic fields and magne-tohydrodynamic (MHD) turbulence generation.Our main astrophysical motivation is to explain the wholevariety of nonthermal and thermal processes occurring at dif-ferent spatial scales of clusters in both central regions, wherethe interaction of active galactic nuclei (AGNs) with the clustermedium is of primary importance, and other places, particularlyat the periphery, with conditions favoring multi-frequency emis-sions.Nonthermal radiation observed from galaxy clusters, whichresults from the interaction of nonthermal relativistic particles,that is, cosmic rays (CRs), with the other components of the cos-mic plasma of the ICM, has built-in characteristic features thatare specific to a given acceleration mechanism. The di ff usivesynchrotron radiation that we observe in clusters is a manifes- tation of the di ff usive Fermi-type acceleration regime, as in thevast majority of nonthermal radiation emitted by di ff erent astro-physical sources. Thus, this is the basic motif of astrophysicalnature, which also indicates galaxy clusters as relevant objectswith which to test the Fermi acceleration mechanism of the firstorder. Features of the nonthermal di ff usion emission produced inthe ICM, in particular its deviation from a simple, so-called uni-versal power-law spectrum, provides the basis for verificationof the DSA model in this environment. In light of recent ob-servations, radio relics are particularly good candidates for suchverification procedures (e.g., van Weeren et al. 2019, Brüggen& Vazza 2020, Bykov et al. 2019). However, a multi-messengerapproach used to study relics in both the radio and X-ray do-mains shows significant discrepancies, especially in estimationsof shock-wave characteristics (Ogrean et al. 2013; van Weerenet al. 2016).Di ff usive radio emission of cluster relics occurring typi-cally on the edges of the clusters of galaxies is associated, be-cause of the lack of an obvious optical counterpart, with theICM environment and indicates the presence of relativistic elec-trons with γ ≈ which are being accelerated within the hotT ≈ −
10 keV (where T is temperature) and magnetized ≈ µ Gplasma. On the other hand, there is a statistical correlation be-tween the global features of relics, including X-ray and ra-dio brightness, and the dynamic state of the cluster (see, e.g.Enßlin & Brüggen 2002, Golovich et al. 2016, Golovich et al.2017, Golovich et al. 2019a and Golovich et al. 2019b). Andvice versa: only in dynamically disturbed clusters (current shock
Article number, page 1 of 8 a r X i v : . [ a s t r o - ph . H E ] F e b & A proofs: manuscript no. paper waves) can one observe the di ff use radio emission in the formof a relic (van Weeren et al. 2019). Therefore, particularly forradio relics, where there is no controversy as to the relationshipbetween the existence of a shock wave and a radio signal, reason-able verification or clarification of the DSA acceleration modelcan be a ff orded.In practice, our aim is to obtain a consistent interpretationof the shock-wave features, which is usually identified witha merger shock, seen in observations in both the radio andX-ray domains. A good, dimensionless parameter characteriz-ing the power of a particular shock is the sonic Mach numberM = v sh / v sound . In the linear version of the DSA model, thisnumber entirely determines the form of the spectrum of accel-erated particles. Therefore, in a first approximation, the agree-ment obtained between two independent Mach numbers, that is,the number resulting from X-ray observations and that result-ing from the radio spectrum, should be su ffi cient confirmationof the proper description of the di ff usive radio emission of therelic by the DSA model (Finoguenov et al. 2010; Akamatsu et al.2012; Sarazin et al. 2016). However, the comparison of relativis-tic electron features seen through the radio observations with theX-ray shock diagnostics implied by thermal plasma fraction re-veals many discrepancies in the estimated shock characteristics.The list of the most common incompatibilities is summarizedby the following problems: – The Mach numbers resulting from the X-ray emission oftendisagree with (are smaller than) those resulting from the ra-dio spectral index (Stroe et al. 2014a,b; van Weeren et al.2016; Hoang et al. 2017; Akamatsu et al. 2017; Ha et al.2018). – The relics radio spectrum often reveals deviations from thepower-law spectrum (Trasatti et al. 2015; Stroe et al. 2016;Malu et al. 2016; Kierdorf et al. 2017; Rajpurohit et al.2019). – Sometimes one can see a spatial o ff set in the locations ofX-ray and radio shocks (Ogrean et al. 2013). – There is a lack of consistency between the deduced Machnumber and the observed suitable electron acceleration e ffi -ciency (Kang & Ryu 2013; Botteon et al. 2020; Lee et al.2020).In the literature, there has been a wide spectrum of ideasput forward to explain these inconsistencies. Most of the con-cepts explaining the problem of the above discrepancies requirethe presence of a residual electron population re-accelerated viaDSA or refer to consideration of significant shock turbulence orthe involvement of adiabatic compression of plasma lobes fromold radio sources. Unfortunately, even these varied approachesdo not comprehensively solve all the problems listed above.In this article, we refer to the idea put forward by Siemieniec-Ozieblo & Golda (2016), according to which the phenomenon ofradio relics could be the result of the simultaneous presence oftwo shock waves with di ff erent Mach numbers; one being a for-ward merger shock approaching the other, the two with defini-tively di ff erent characteristics, for example, internal accretionshock or inflow shock. We expect that within such a scenario,the DSA acceleration process will be modified by the presenceof this latter shock. In particular, the resulting morphology of ra-dio emission, including the flattening or steepening of relic spec-trum observed in certain cases, may be the property linked tothe mutual influence of two shocks situated near to one another.Such a two-shock configuration has seldom been discussed in theliterature. However, several papers have recently appeared that consider a two-shock scenario in the context of either magneto-sphere or solar environment (e.g., Hietala et al. 2011). Moreover,Stroe et al. (2014b) show that a single power law does not accu-rately describe the spectrum of di ff erent parts of radio relics inthe Toothbrush and Sausage clusters. Stroe et al. (2016) suggesta broken power-law model to describe the spectral steepeningaround 2-2.5 GHz of each of the relics. We believe that the obser-vations of 1RXS J0603.3 + + ff erent dif-fusivity and thus might explain the subtle features at the electronemission spectrum at 200 −
300 MHz visible in both observationsand simulations.We focus here on the parameters that may be of significantimportance with respect to changes in the profile of the electronspectrum. We describe the parameters, vary them, and try to ex-plain the trend of the concave shape that appears in the spectrumof cosmic ray electrons (CRe) and is revealed in both observa-tions and simulations of shock waves in galaxy clusters (e.g.,Basu et al. 2016 and Nakanotani et al. 2018). To do this we de-rive a new solution for the adequate electron transport equationand analyze the influence of some of the most important dimen-sionless parameters on the shape of the spectrum. We also ex-plicitly show that the fossil electrons swept away by the innershock could be one of the causes of the occurrence of spectralshape distortions. We show in general that the CRe spectrum isvery susceptible to dependence on certain parameters, contraryto what is expected in pure DSA.Thus, the purpose of this paper is to show that a double-shockbuild up with one merger shock and an invisible accretion (virial)shock can change the so-called universal CR spectrum even inthe linear regime of acceleration. The outline of the article is asfollows: in Sect. 2 we present the mathematical model that werely on to obtain the results which we describe and discuss inSect. 3. The results section is followed by a summary.
2. Theoretical approach
Cosmic-ray acceleration in a system consisting of two relativelyclose shocks was recently analyzed in the literature, mainly inthe context of interplanetary shocks (e.g., Hietala et al. 2011).Both the empirical spectra of detected ions and the results ofparticle simulations showed that in the case of two shocks, thedescription of the acceleration process via DSA in terms onlyof a single shock does not reproduce the observed results. Thespectral index of radio emission for a pair of close shocks turnedout to be harder than that of a single shock.In a recent paper, Nakanotani et al. (2018) discussed the re-sults of a test-particle simulation describing particle accelerationin the presence of two shocks, which finally generated a bentspectrum of particles in the form of a double power law. Here, inthe case of a radio relic we also examine the DSA process in asystem including two shocks. In the current section we describethe theoretical model that we use in our calculations.A merger shock wave propagating outward from the clusteris considered. The inflowing gas also o ff ers an additional mecha-nism for shock formation within a galaxy cluster volume, which Article number, page 2 of 8razyna Siemieniec–Ozieblo and Mariia Bilinska: Acceleration of cosmic rays by double shock waves in galaxy clusters: application to radiorelics takes place both at the cluster border and within the whole vol-ume of a galaxy cluster. Such an inflowing shock (hereafter re-ferred to as an "accretion" shock) has a lifetime scale compara-ble to the Hubble time, while the lives of the merger shocks areof the order of the cluster dynamical timescale. Accretion shockallows matter to be accreted along the filaments into the ICM.In the current paper we consider a system of two such shocks:an accretion shock sh sh
2. As the accretionshock is formed due to the inflow of matter from outside the clus-ter, it is stronger than the merger shock, that is, its Mach numberis larger than the merger shock Mach number ( ≥ ∼
2) (Honget al. 2015). Compared to the merger shock, the accretion shockcan be considered as stationary; moreover, in our considerationwe treat both shocks as not moving with respect to each other,which means that the timescale of acceleration is less than thetime needed for the merger shock to overlap the accretion shock.The position of the radio relic should roughly coincide with themerger shock. In our analysis we use the di ff usion convectionequation for three regions defined with respect to the positionsof the two shocks (see Fig. 1): an absolute downstream region,an inter-shock region (between the two shocks), and an absoluteupstream region. Fig. 1.
Schematic image of a one-dimensional double-shock sys-tem. The image shows the directions of two shock waves, theflow speeds, the position of a radio relic, and the three shockregions.In each of the shock-related regions, values of di ff usion co-e ffi cients κ i , flow velocity u i , and particle distribution function f i ( x , p ) ( i = , , , , ff erent. We consider the injection to be sta-tionary which makes the injection terms Q and Q constant.As they spread towards the outskirts of the galaxy cluster, thepresence of the shock waves leads to a decrease in the flow veloc-ity in the directions of the cluster center; hence u > u > u and x , x denote the positions of the shocks; furthermore (as seenin Fig.1), for simplicity we set x =
0. The above parametersare the main transport parameters of the system. Additionally,we consider compressions r = u / u and r = u / u which areratios of flow velocities.In our consideration, both merger and accretion shocks areinfinite and planar. However, these meet di ff erent mechanismsof injection: for the outer accretion shock it is a supra-thermalseed accompanied by a marginally small number of fossil elec-trons that could be able to di ff usively reach that shock, whilefor the inner merger shock the presence of mildly relativisticelectrons is crucial. The fossil population is assumed, for sim- plicity, to consist of mono-energetic (not power-law) electronsinjected by AGNs to the ICM, cooled, and then swept away bymerger shocks. The injection mechanisms still require a betterunderstanding and detailed description. Nevertheless, recent re-sults from simulations of the collision-less shocks show that thefraction of injected particles is a function of many parameters(e.g., Caprioli et al. 2018). Therefore, it is impossible to quan-tify the Q i values in general terms. In the following, we use thenondimensional parameter Q / Q to represent the ratio of bothinjections. To estimate the range of the ratio of the injection am-plitudes Q / Q , we roughly assume the proportionality relationto the particle densities injected into the acceleration process,i.e., Q / Q ∝ n e , / n e th , where n e , changes from n e th , in case ofsupra-thermal injection at the merger shock up to n e rel , for in-jection occurring in the lobe environment which could be placedthere. Here, n e is the number density of injected electrons; n e th are thermal electrons and n e rel are relativistic. Due to the largergas density at shock 2 and more e ffi cient injection in the caseof the compressed (e.g., radio galaxy) lobe, we expect that thedi ff erence between the number of particles injected into bothshocks may become essential (see Sect. 3.1) for the acceleratione ffi ciency of this double-shock system.Therefore, considering a rather simple model and focusingmainly on the role of injection in this dual-shock system, we con-sider the di ff usion coe ffi cients to be independent of the energyof the particles, but di ff erent in each of the regions: κ i (cid:44) κ i ( p ).Otherwise, the results of a di ff usion-based mechanism of ac-celeration become dependent on the "momentum scale". If thedi ff usion coe ffi cient were to show a dependence on momen-tum (i.e., λ ( p )), one might expect the kind of "spectral break-ing" feature. This could be produced because of the momentumdependence in later discussed, dimensionless parameter: shockdistance / di ff usion length(p). This shows that only particles withhigher momenta than the "breaking" point value would be accel-erated in both shocks. To avoid the description of such a complexscenario we restrict ourselves to the above-described, unrealisti-cally simple form of di ff usion coe ffi cient.In such a system of two relatively close shocks, particle dis-tribution is governed by a di ff usion-convection equation. The CRtransport equation for the space distribution function is as fol-lows: u i ( x ) ∂ f i ( x , p ) ∂ x − κ i ( x ) ∂ f i ( x , p ) ∂ x − ∂ u i ( x ) ∂ x p ∂ f i ( x , p ) ∂ p − Q i ( x , p ) = , (1)where f ( x , p ) is the coordinate- and momentum-dependentdistribution function; u i is the velocity of inflowing matter( u > u > u ); κ i ( x ) is the di ff usion coe ffi cient; andQ i (x , p) = Q i δ (x − x i ) δ (p − p ) is the source term at the posi-tion of each of the two shocks.Below, we assume flow velocities and di ff usion coe ffi cientsin all three regions to be constant for each region, i.e., u i (cid:44) u i ( x )and κ i (cid:44) κ i ( x ) . As discussed above, we consider a stationaryscenario with f i (cid:44) f i ( t ) . Under these assumptions we are searching for the separablesolution, f ( x , p ) = g ( x ) h ( p ) , of eq. (1), leading to the generalsolution of the form (for i = , f i ( x , p ) = κ i u i A i ( p ) exp (cid:32) − u i κ i x (cid:33) + B i ( p ) . (2) Article number, page 3 of 8 & A proofs: manuscript no. paper
Boundary conditions determine whether this separable solu-tion is relevant or not. Below we postulate the physically reason-able boundary conditions: • f ( ∞ , p ) = • f ( x , p ) = f ( x , p ) and f ( x , p ) = f ( x , p ); • Q (x , p) = [S] x and Q (x , p) = [S] x ; • f ( −∞ , p ) < ∞ . Here, S i ( x , p ) ≡ − π p (cid:32) u i p ∂ f i ∂ p + κ i ∂ f i ∂ x (cid:33) is the di ff erential par-ticle flux and [ S ] x i = S i ( x , p ) | x > x i − S i ( x , p ) | x < x i represents thedi ff erence between fluxes on both sides of the shock.Above we have applied the continuity requirement for parti-cle distribution functions and di ff erential particle fluxes, where Q represents the preexisting (fossil) mildly relativistic electronsinjected for re-acceleration when the merger shock crosses theirlocation on its way to approach the accretion shock.Using the above conditions, first we reduce the problem offinding six functions A i ( p ) , B i ( p ) into the coupled system of twofirst-order di ff erential equations for A ( p ) , A ( p ) , B ( p ) , B ( p )(because A ( p ) = = B ( p )). Thus, we are finally able to obtainthe ultimate second-order decoupled equation for either A ( p )or A ( p ) which has the form of a linear Euler-type equation.The eliminated remaining functions can easily be calculated af-ter solving the latter equation. The general form of a solution forany of these functions has the form B i ( p ) , A i ( p ) = C i · p α + C i · p α , (3)where negative spectral indices α , are very complicated alge-braic functions of many parameters ( x , κ , u , r and r ). Theexpression for α , is presented below. α , = · (cid:34) + l · exp (k x )exp (k x ) − ± (cid:113) exp (k x ) · (exp (k x ) + · ( l + l ) + x ) − (cid:35) , (4)where k is given below after eq. ( 5) and l = r (1 − r ) + r ( r − r − , l = r − r ( r − r − , l = r + r ( r − r − . A typical functional behavior of these indexes is shown inFig. 2.Below we restrict our concern to region 2, where particlescan potentially participate in acceleration in both shocks. Oncethe solution of the Euler-type equation for A is found and B is immediately calculated, one can find the expression for thedistribution function f ( x , p ) in the inter-shock region, which isgiven below: f ( x , p ) = k (cid:34) exp [(k − k )x − k x] (cid:32) ξ + r − r η (cid:33) κ κ − exp (k x ) (cid:32) ξ + η (cid:33) ( r − (cid:35) , (5) Fig. 2.
Dependence of the α i parameters on the di ff usion coe ffi -cient of the inter-shock region κ (green line: α , red line: α ).where k i = u i /κ i ,ξ = p α ( C − G ) + p α ( C + H ) ,η = α p α ( C − G ) + α p α ( C + H ) , C and C are two arbitrary constants, G = exp k x p − (1 + α )0 (3 Q r + Q (3 r exp k x )) κ l (cid:113) (exp k x + l + l ) + ++ α (exp k x − r − κ l (cid:113) (exp k x + l + l ) + , H = exp k x p − (1 + α )0 (3 Q r + Q (3 r exp k x )) κ l (cid:113) (exp k x + l + l ) + ++ α (exp k x − r − κ l (cid:113) (exp k x + l + l ) + , l = ( r − r − r . The main purpose of this section is to infer information con-cerning the structure of the solution and to highlight its depen-dence on important physical parameters. As seen in Eqs. (3)-(5),the distribution function is expressed as a combination of twopower-law functions whose indexes α i depend on the compres-sion r i of both shocks and the dimensionless parameter u x /κ (which we comment on below). However, in the case of the com-bined action of the two shocks, the general shape of f ( x , p ) isalso modeled by the additional parameter of relative injection Q / Q . Therefore, in addition to compression, injection explic-itly a ff ects the form of the particle distribution as well, and thespectrum of particles and their synchrotron emission will inheritthese influences. Therefore, the interpretation of possible spec-tral distortions, in particular those found in the lower energyrange, should be more distinct and unambiguous in this area be-cause of the lack of significant radiative losses. Article number, page 4 of 8razyna Siemieniec–Ozieblo and Mariia Bilinska: Acceleration of cosmic rays by double shock waves in galaxy clusters: application to radiorelics
In this article, we limit ourselves by analyzing the distribu-tion function of particles as a function of x and p . Also, we re-veal the impact on its profile made by dimensionless parameters r , r , κ /κ , Q / Q , and u x /κ .The latter parameter a ff ects the distribution function not onlythrough the coe ffi cients in the combination of two power lawsbut also by the α i spectral indexes. It is worth noting that thenumerical values of u x /κ = L /λ , which is equal to the ra-tio of shock distance over di ff usion length. This introduces, onthe other hand, a constraint on the permissible κ value, whichensures a negative value of α ( α is always negative). The ex-pected negative values of both α i are the result of an asymptoticsolution in the case of large x . In order to regain, in the extrap-olation process, the appropriate power-law behavior the shocksmust be su ffi ciently separated, that is, in the case of two indepen-dent acceleration processes, one has to obtain power-law spectra ∝ p α i , where α i <
0. It can be shown that for x ≥
10 kpc and forrealistic values of κ ≤ cm / s , α is negative, which resultsfrom the fact that the factor e k x = e u x /κ (cid:29) κ / u ≤ x = L , and therefore equivalentto the condition of the possible acceleration by both shocks. Inother words, depending on the distance between the shocks andthe value of κ , a certain number of particles can reach the ac-cretion shock, and so both shocks take part in the accelerationprocesses. We take the value of the inter-shock region width sothat an actual double shock system is modeled: if the distance istoo great, there is no way the particles can interact with the twoshock waves, which simply leads to two separate shock accel-erations; if x is too small, it is as if only one single shock wasoperating as accelerator. As a consequence, this can introduce adistortion from a pure power law to a distribution function andsubsequently to the electron spectrum.
3. Results
In this section we discuss the results obtained on the basis of ourmodel. Among the three regions appearing in the double-shockmodel, the inter-shock region is the most interesting: in betweentwo shocks we expect a higher level of turbulence, situated ina collapsing trap of the inter-shock region (see Hietala et al.2011).From the theoretical model described in Section 2, we obtaina complicated distribution function f ( x , p ) depending on manyparameters (distance between two shocks, compression ratios,spatial di ff usion coe ffi cients, injection values, etc.). Due to thenumber of parameters that are to be fit, we introduce additionalassumptions. As the accretion shock is supposed to be strongerthan the merger shock (as shown in Hong et al. 2015), here weassume r / r <
1. In particular, we present the case r = r = .
5. Based on the assumptions from Hong et al. (2015),we take the velocity of the upstream flux to be equal to u ≈ m / s. We take di ff usion coe ffi cients so that κ > κ (due to theincreased level of turbulence in region 2) and the source term Q / Q (cid:29)
1, because the injection of the internal region is ex-pected to be much stronger due to the fact that any nonthermalsources inside the system of merging clusters can provide an ad-ditional population of mildly relativistic particles (e.g., AGNs).It is important to note that the spectrum we expect to see fromour analysis is not a power-law ‘straight’ line predicted by DSA,but a bent spectrum, signs of which can be seen both in the ob-servations (see, e.g., Fig. 10 in Basu et al. 2016) and in the re-sults of simulations (Nakanotani et al. 2018). At high frequen-cies, such an e ff ect may be explained by the influence of theSunyaev-Zeldovich e ff ect (Basu et al. 2016) plus losses and the acceleration processes themselves (e.g., Shock Drift Accelera-tion mechanism, see Matsukiyo et al. 2011) on the spectrum.Here, we focus on lower frequency spectral peculiarities. Oneof the reasons for such peculiarities at low frequencies may bean impact of the accretion shock wave. Below we show the wayin which each parameter influences this deviation. We expect theconcave spectrum when fitting the parameters Q , Q , κ , κ and x . Below we quantify or discuss values of κ i , u i , and Q i . Here α i ≡ α ( r i , u i , κ ); i = ,
2. Spectral indexes are given by eq.4, α is always negative for any reasonable values of r , r , u , κ whilst α is negative at particular x only at a chosen range of κ . We take x ≥
10 kpc, and in our choice of very small x we are mostly governed by our interest to consider a system oftwo shocks shortly before their collision (for a post-collision sce-nario, see Zhang et al. (2020a,b).As we consider the inter-shock region to encompass the mostinteresting physical processes among the other regions, it is the f ( x , p ) part of the distribution function that we concentrate on.For simplicity, by default we set C = C = C i are being discussed. Below, we discuss eachparameter in turn, thus dividing our analysis into a number ofsections. At closer distances to the galaxy cluster center we assume anadditional contribution of particle populations from inside thegalaxy cluster (e.g., AGNs). For this reason, the expected rela-tion between the source terms is Q / Q >
1. Figure 3 showsthat the distribution function f ( x , p ) shows a significant depen-dence on the momentum when Q / Q (cid:29)
1. As an example,here we compare a particular distribution function f ( x , p ) forseveral values of the ratio Q / Q covering the di ff erent values of n e , . The series of curves show the whole range of injection pa-rameter Q starting from the values characterizing supra-thermalelectrons where particle density in a typical ICM environmentfor di ff erent shock locations is equal to n g as , ∼ − cm − , upto n g as , ∼ − cm − (thus injected n e th , i fractions are smallerby about 10 times; see, e.g., Ksenofontov & Berezhko 2018).While for mildly relativistic fossil plasma we expect much moree ffi cient injection resulting in an electron number density of n e , ≡ n e rel , ∼ − cm − . It is interesting to mention that in caseswhere Q / Q ≤ , at low energies the shape of the curve maychange from concave to convex. We set di ff usion coe ffi cients as constant throughout the corre-sponding regions. In this section, two types of plot are shown: f ( x , p ) function with respect to κ and κ (see Fig. 4). The valuesof di ff usion coe ffi cients κ i are adopted according to the scalingformula (see, e.g., Berezinsky et al. 1997) which for the parti-cles with γ ∝ and for the decreasing intergalactic magneticfield with x down to the order of B ∼ κ ∼ · cm s − and take the slightly lower valuefor κ .As we consider an inter-shock region as a collapsing trap,the inequality κ > κ should be fulfilled. Here, the κ ∼ κ curve has a di ff erent shape from κ /κ ≥ . κ /κ ∝ . x , the larger the distance between the spectral curves with Article number, page 5 of 8 & A proofs: manuscript no. paper
Fig. 3.
Dependence of the distribution function of the inter-shockregion f ( x , p ) at x =
10 kpc, for Q / Q = , . , , , , × on CR momentum.respect to κ . Here, f ( x , p ) is approximately of the same orderas where there is a small distance between the two shocks. Fig. 4.
Dependence of the distribution function on the CR mo-mentum for di ff erent di ff usion coe ffi cients. Top panel:
As a func-tion of varying di ff usion coe ffi cient of the inter-shock region. Bottom panel:
As a function of varying di ff usion coe ffi cient ofthe absolute upstream. We note the dashed ("saturation") curve.It must also be noted that variation of κ reveals the ex-istence of a limiting (“saturation”) curve of inter-shock distri-bution function f ( x , p ). In the case of very small x , whichis of the order of 10 kpc, the saturation curve occurs at κ = m / s which constrains the κ coe ffi cient range to 10 . m / s to 10 m / s at κ = m / s (Fig. 4, lower panel). In order to understand all the e ff ects manifesting in Fig. 4 itis important to note that we have taken a relatively large value of κ here, which nevertheless still enables acceleration with par-ticipation of both shocks. For this chosen value of κ , the dis-played saturation e ff ect with respect to κ reveals the physicalupper limit for particle acceleration. This means that for largervalues of κ , the particles meet the upper bound to return to shock1. Larger values of κ , which means larger χ ( χ ∝ B / ( δ B ) indi ff usion coe ffi cient), means much weaker scattering and thusless e ffi cient acceleration. In other words, for these larger val-ues of κ we have to take particle escape into account. The in-creasing gap (not shown here) in (e ffi ciency) normalization withshock distance is due to the decreasing number of particles thattake part in acceleration. Here, di ff usion length becomes muchsmaller than the distance that must be reached by particles in or-der for them to be accelerated in both shocks. On the contrary,the thicker curve shows that the ‘double’ shock is more e ffi cientat comparable and relatively large values of both di ff usion coef-ficients. Thus, comparable mean free paths at both sides of shock1 lead to stronger scattering in the whole region, which leads toa flatter spectrum and greater acceleration e ffi ciency. The particular value of x has almost no influence on the profileof the spectrum curve. This means that the position we take inthe range of x > x > Dependence of f ( x , p ) on compression in the inter-shock region r always reveals an expected bent behavior at r < r =
4. It is interesting that in the case of r = . , the curve has a deeper bending than those with greater values ofcompression. Increasing the x distance up to 200 kpc leads to avisibly di ff erent behavior of the distribution function (see Fig. 6)with a convex spectrum peculiarity. As for the case of the equalcompressions, r = r = . , . , .
75, the curve correspondingto r = r = .
25 is steeper than the others (Fig. 7).
Fig. 5.
Dependence of the distribution function on the CR mo-mentum for di ff erent values of compression r at the position ofthe second shock, r = ff usion coe ffi cients in the absolute upstreamand inter-shock regions as well as the injection terms in the sameregions, play an ultimate role in the formation of the disturbedspectrum of CRe observed at 200 −
300 MHz.
Article number, page 6 of 8razyna Siemieniec–Ozieblo and Mariia Bilinska: Acceleration of cosmic rays by double shock waves in galaxy clusters: application to radiorelics
Fig. 6.
Dependence of the distribution function on the CR mo-mentum for di ff erent values of compression, r and r , at theposition of the second shock at 200 kpc. Fig. 7.
Dependence of the distribution function on the CR mo-mentum for r = r = r compression at the position of thesecond shock at 200 kpc.
4. Summary Di ff use synchrotron emission radiating from many astrophysicalobjects sometimes reveals a spectral deviation from a power-lawspectrum and we believe that the mechanism behind the repre-sentation of this phenomenon should rely on Fermi I acceler-ation. In the present paper, we concentrate on radio relics, assuch objects seem to be a straightforward example of the FermiI mechanism.In the current study, we consider a magnetic field strength ina cluster ICM of around 1 µ G (see, e.g., Section 4 in van Weerenet al. 2019). In terms of such an assumption, the emission seenat a few hundred MHz is produced by electrons with energies ofa few GeV. We believe that any additional features manifested inthe shape of the radio emission spectrum are a reflection of thevariation in the momentum slope of the electron spectrum.The deviations in the flux of radio emission may happen ata di ff erent frequency range. For instance, observations of theToothbrush and Sausage relics show a clear deviation from apower law observed in the form of a concave shape at low fre-quencies of around 200 −
300 MHz, while at slightly higher fre-quencies of 1 − . ff ect (Basu et al. 2016). Conversely, wefocus on subtle deviations at about 200 −
300 MHz, which tothe best of our knowledge have not yet been discussed in lit-erature. We suppose that at the position of a radio relic such afeature is produced by the presence of two shocks. Nakanotaniet al. (2018) present simulations that support our consideration.We believe that the deviation at low frequencies of ∼ − ∼ µ G for low-energy elec-trons. The concept of two shocks also shows promise for explain-ing the Mach number and other discrepancies (see, e.g., Ogreanet al. 2013 and van Weeren et al. 2016). Nevertheless, the morefar-reaching motive to examine this is to make the DSA modelconclusive at low frequencies. We believe that as a consequence,such consideration of the low- and high-frequency parts of thespectrum may allow the whole radio emission distribution to beproperly and consistently described.In the present paper we study a distribution function of elec-trons and speculate that the characteristics of its shape could beimprinted in the detected radio spectrum. We analyze the ex-istence of both bumpy or concave features in the model of adouble shock wave. Also, we presume that for the two shocksconsidered in our model (a weaker merger shock and a strongeraccretion shock) the origin of injection may di ff er. We assumethat a (internal) merger shock has an additional injection of aCR population from the galaxy cluster volume (AGN influence),while the (outer) accretion shock receives injected particles fromthe inflow of matter from outside the galaxy cluster. The dou-ble shock-wave system suggests the existence of three regions:an absolute upstream region (upstream for accretion shock), aninter-shock region (which is downstream from the accretion andupstream from the merger shock), and an absolute downstreamregion (downstream from the merger shock, instead of upstreamand downstream as in the case of one single shock). We ex-pect a higher level of turbulence of the CRs in the inter-shockmedium, which means that this region roughly overlaps the relic.The increased turbulence makes this region a kind of collaps-ing trap (Hietala et al. 2011) in which the rate of accelerationincreases with decreasing distance between the shocks. As theinner merger shock approaches the outer accretion shock, thedistance between them decreases making the inter-shock regionnarrower. According to Hietala et al. (2011), if the second adia-batic invariant remains constant, the parallel momentum of par-ticles trapped between the shocks should increase, increasing theacceleration rate. We believe that, at each region, mechanisms ofdeviation from the power law are due to di ff erent phenomena.In the present paper we show that the inter-shock regionwidth of about the width of a relic reveals an expected concavespectrum peculiarity, while the increasing distance leads to theappearance of a convex curve. We consider a solution in terms ofa transport equation with initial conditions that involve a popula-tion of fossil electrons, expressing continuity of both the di ff er-ential particle flux S ( x , p ) and the electron distribution function f ( x , p ) at the positions of the two shocks. In other words, westate that the distribution function or particle flux on both sidesof each contact discontinuity are equal to each other, whilst thedistribution function far from the galaxy cluster f ( ∞ , p ) is equalto zero and the distribution function close to the cluster center f ( −∞ , p ) may be of any finite value. In addition to the above-stated consideration, we use a nonstandard representation of thematching conditions representing the flux. This expresses theconcept that the additional injection to the system is e ff ectuated Article number, page 7 of 8 & A proofs: manuscript no. paper in the case where a weak (merger) shock meets fossil popula-tions of electrons on its way to approaching the accretion shock.We believe that even if a weak shock is not visible, it may ig-nite fossil electrons unevenly distributed in the vicinity of theaccretion shock. However, for a bent spectrum to be revealed,both shocks need to be su ffi ciently close (as we show, down to adistance comparable to the width of a relic). As Q and Q arethe terms that represent the injection process, the dimensionlessrelation Q Q should be of major importance due to the additionalinjection of fossil electrons to the supra-thermal part.We find that the disturbed shape of the spectrum in the inter-shock region mostly depends on the following parameters:a) The ratio of injection. The e ff ect of distortion of the electronspectrum can easily be seen at large ratios of source terms Q Q ∝ ; this conclusion coincides with our expectations asto the importance of the source terms on the spectrum profile.We consider the di ff erence between Q and Q to be the maincontributor to spectral deviation at 200 −
300 MHz.b) Di ff usion coe ffi cients of absolute upstream and inter-shock.Although we consider a rather simplistic model in which dif-fusion coe ffi cients do not depend on energy, even in this casewe obtain a distorted spectrum. We see that the influence of κ i is very important anyway and conclude that the energydependence on the di ff usion coe ffi cient may be neglectedfor our current purposes. Di ff erent ratios of the absolute up-stream κ and inter-shock κ di ff usion coe ffi cients produce adi ff erent behavior of electron spectrum. The spectrum closestto that observed in Basu et al. (2016) and Nakanotani et al.(2018) is obtained at κ κ ∼ . ff usion coe ffi -cient ratios, and in addition to the distance between the twoshocks, the compression also shows an obvious impact onthe profile of the CRe spectrum. At inter-shock distancesof the order of several tens of kiloparsecs, both at low andstrong compression, a convex spectrum peculiarity is ob-served, while distances of the order of the width of the relicreveal a concave shape. Moreover, spectra corresponding tosmaller values of compression are found to be steeper.As expected, the position of the probe between the twoshocks has almost no influence on the profile of the spectrumcurve. For this reason, and for the reasons outlined above, weinterpret our results as suggesting that the physics behind theinjection lead to a much more pronounced e ff ect than the vari-ability in the shock properties.Our results also appear to suggest that the described discrep-ancy may be due to the presence of more than one cotemporaldiscontinuity. As various authors show, the presence of spectraldeviations in the Toothbrush and Sausage relics suggest that ex-pansion of DSA theory at low energies is needed. We believethat a double-shock model introduces a fresh approach to theproblem. In fact, our model that represents a linear double-shockwave with particles in the inter-shock region that “feel” the pres-ence of both waves, is similar to a single nonlinear shock witha concave power-law spectrum (e.g., Fig. 1 in Amato & Blasi2006). We assume that an o ff set of 1-arcmin (which correspondsto 217 kpc in the redshift of the Toothbrush cluster) between theposition of the merger and the unseen accretion shocks in radioand X-ray bands described in Ogrean et al. (2013) is a good il-lustration of a double-shock model. Another argument in favor of a double-shock model may be the existence of a confusingdiscrepancy between the Mach number derived from the spec-tral index and the one estimated from X-ray observations. Kanget al. (2017a) note that a possible explanation to this could lie inthe fact that a radio relic may consist of multiple shocks with dif-ferent strengths. Namely, they state that X-ray observations tendto pick up the parts of shocks with lower Mach numbers andhigher kinetic energy flux, while radio emissions come prefer-entially from the parts of the shocks with higher Mach numbersand higher CR production. Various authors (Hoang et al. 2017;Skillman et al. 2008; Vazza et al. 2009) suppose that the mergershocks responsible for radio relics in galaxy clusters (in LSS for-mation simulations of the Universe) consist of multiple shockswith di ff erent Mach numbers. Acknowledgements.
We thank to the anonymous reviewer for helpful and in-sightful comments and to prof. Zdzisław Golda for highly valuable assistance incalculating the constituent functions of the distribution function. Mariia Bilinskaalso appreciates the NASA / IPAC NED operated by Jet Propulsion Laboratory,Caltech, under contract with the NASA, and NASA ADS operated by the Smith-sonian Astrophysical Observatory under NASA Cooperative Agreement.
References